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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 05 Nov 2024 06:55:43 -0800A Note on the Helical Movement of Micro-Organisms
https://resolver.caltech.edu/CaltechAUTHORS:20150211-145731256
Year: 1971
DOI: 10.1098/rspb.1971.0068
This note seeks to evaluate the self-propulsion of a micro-organism, in a viscous fluid, by sending a helical wave down its flagellated tail. An explanation is provided to resolve the paradoxical phenomenon that a micro-organism can roll about its longitudinal axis without passing bending waves along its tail (Rothschild 1961, 1962; Bishop 1958; Gray 1962). The effort made by tho organism in so doing is not torsion, but bending simultaneously in two mutually perpendicular planes. The mechanical model of the micro-organism adopted for the present study consists of a
spherical head of radius ɑ and a long cylindrical tail of cross-sectional radius b, along which a helical wave progresses distally. Under the equilibrium condition at a constant forward speed, both the net force and net torque acting on the organism are required to vanish, yielding
two equations for the velocity of propulsion, U, and the induced angular velocity, Ω, of the organism. In order that this type of motion can be realized, it is necessary for the head of the organism to exceed a certain critical size, and some amount of body rotation is inevitable. In fact, there exists 1m optimum head-tail ratio ɑ/bat which the propulsion velocity U reaches a maximum, holding the other physical parameters fixed. The power required for propulsion by means of helical waves is determined, based on which a hydromechanical efficiency η is defined. When the head-tail ratio ɑ/b assumes its optimum value and when
b is very small compared with the wavelength λ, η ≃ Ω/ω approximately (Ω being the induced angular velocity of the head, ω the circular frequency of the helical wave). This η
reaches a maximum at kh ≃ 0.9 (k being the wavenumber 2π/λ, and h the amplitude of the helical wave). In the neighbourhood of kh = 0.9, the optimum head-tail ratio varies in the range 15 < a/b < 40, the propulsion velocity in 0.08 < U/c < 0.2 (c = ω/k being the wave phase velocity), and the efficiency in 0.14 < η < 0.24, as kb varies over 0.03 < kb < 0.2, a range of practical interest. Furthermore, a comparison between the advantageous features
of planar and helical waves, relative to each other, is made in terms of their propulsive velocities and power consumptions.https://resolver.caltech.edu/CaltechAUTHORS:20150211-145731256Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies
https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm74
Year: 1974
The present series of studies is concerned with low-Reynolds-number flow in general; the main objective is to develop an effective method of solution for arbitrary body shapes. In this first part, consideration is given to the viscous flow generated by pure rotation of an axisymmetric body having an arbitrary prolate form, the inertia forces being assumed to have a negligible effect on the flow. The method of solution explored here is based on a spatial distribution of singular torques, called rotlets, by which the rotational motion of a given body can be represented.
Exact solutions are determined in closed form for a number of body shapes, including the dumbbell profile, elongated rods and some prolate forms. In the special case of prolate spheroids, the present exact solution agrees with that of Jeffery (1922), this being one of very few cases where previous exact solutions are available for comparison. The velocity field and the total torque are derived, and their salient features discussed for several representative and limiting cases. The moment coefficient C[sub]M = M/(8[pi][mu][omega sub 0]ab^2) (M being the torque of an
axisymmetric body of length 2a and maximum radius b rotating at angular velocity [omega], about its axis in a fluid of viscosity [mu]) of various body shapes so far investigated is found to lie between 2/3 and 1, usually very near unity for not extremely slender bodies.
For slender bodies, an asymptotic relationship is found between the nose curvature and the rotlet strength near the end of its axial distribution. It is also found that the theory, when applied to slender bodies, remains valid at higher Reynolds numbers than was originally intended, so long as they are small compared with the (large) aspect ratio of the body, before the inertia effects become significant.https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm74Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows
https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm75
Year: 1975
The present study furthcr explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they
include the Stokeson and its derivatives, called the roton and stresson.
These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyperbolic
profiles), while the body shapcs cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm75Hydromechanics of low-Reynolds-number flow. Part 4. Translation of spheroids
https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm76
Year: 1976
The problem of a uniform transverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers has been analysed by the method of matched asymptotic expansions. The solution is found to depend on two Reynolds numbers, one based on the semi-minor axis b, R[sub]b = Ub/v, and the other on the semi-major axis a, R[sub]a = Ua/v (U being the free-stream velocity at infinity, which is perpendicular to the major axis of the spheroid, and v the kinematic viscosity of the fluid). A drag formula is obtained for small values of R[sub]b and arbitrary values of R[sub]a. When R[sub]a is also small, the present drag formula reduces to the Oberbeck (1876) result for Stokes flow past a spheroid, and it gives the Oseen (1910) drag for an infinitely long cylinder when R[sub]a tends to infinity. This result thus provides a clear physical picture and explanation of the 'Stokes paradox' known in viscous flow theory.https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm76Hydrodynamic pressures on sloping dams during earthquakes. Part 2. Exact theory
https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm78b
Year: 1978
DOI: 10.1017/S0022112078001640
The equations for the earthquake forces on a rigid dam with an inclined upstream face of constant slope are solved exactly by two-dimensional potential-flow theory. The distribution of the hydrodynamic pressure along the upstream face and the total horizontal, vertical and normal loads on the dam are computed from the integral solutions. The results obtained from the exact theory are compared with those derived from the momentum-balance method and there is reasonable agreement.https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm78bHydrodynamic pressures on sloping dams during earthquakes. Part 1. Momentum method
https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm78a
Year: 1978
DOI: 10.1017/S0022112078001639
Von Kármán's momentum-balance method is adopted to investigate the earthquake forces on a rigid dam with an inclined upstream face of constant slope. The distribution of the hydrodynamic pressure along the upstream face is determined. It is found that the maximum hydrodynamic pressure occurs at the base of the dam for any inclination angle between 0 and 90°. Explicit analytical formulae for evaluating the total horizontal, vertical and normal loads are presented and a useful approximate rule for practical engineers is given.https://resolver.caltech.edu/CaltechAUTHORS:CHWjfm78a